Earliest Known Uses of Some of the Words of Mathematics (Q)

Last revision: July 7, 2017

Q. E. D. In the Elements Euclid concluded his proofs with
ὅπερ ἔδει δεῖξαι
“that which was to be shown”: see e.g. the end
of the proof of Proposition 4 on p. 11 of Fitzpatrick’s
Greek-English Euclid. Medieval
geometers translated the expression as quod
erat demonstrandum (“that which was to be proven”).

According to Veronika Oberparleiter, the earliest known use in print
of the phrase quod erat demonstrandum in a Euclid translation
appears in the translation by Bartholemew Zamberti published in Venice in 1505.

In 1665 Benedictus de Spinoza (1632-1677) wrote a treatise on ethics,
Ethica More Geometrico Demonstrata, in which he proved various
moral propositions in a geometric manner. He wrote the abbreviation
Q. E. D., as a seal upon his proof of each ethical proposition.

QUADRANGLE is found in the geometry sense in English in about 1398 in
a translation of Bartholomaeus Anglicus’ De Proprietatibus Rerum. [OED]

The word was used by Shakespeare, in the sense of
a rectangular space or courtyard.

QUADRANT, as one fourth of a circle, is found in 1571 in Pantometria by Digges:
"Biv, A Quadrant is the fourth part of a Circle, included with two Semidiameters" (OED).

In 1810 Trigonometry, Plane and Spherical; with The Construction and
Application of Logarithms by Thomas Simpson has:
"The difference of any arch from 90 degrees (or a quadrant) is called its
compliment; and its difference from 180 degrees (or a semicircle) its
supplement." [Ryan M. Castle]

In 1849, Trigonometry and Double Algebra by August de Morgan has:
"The Axes divide the plane into four quarters:
and as a line, revolving positively, passes from 0 to 90°,
from 90° to 180°, from 180° to 270° , and from 270° to 360°, it is
said to be in the first, second, third, and fourth quarters of space.
But these might equally well be designated as the ++, +-, --, and -+
quarters of space." [Pat Ballew]

In 1895, Plane and
Spherical Trigonometry, Surveying, and Tables by George A Wentworth has:
"Let, also, the four quadrants into which the circle is divided by the horizontal and
vertical diameters, A A’s, and B B’s, be numbered I, II, III, IV, in the
direction of motion." [Pat Ballew, Univ. of Mich. Digital Library]

QUADRATFREI. See the entry SQUAREFREE or SQUARE-FREE.

QUADRATIC is derived from the Latin quadratus, meaning
"square." In English, quadratic was used in 1668 by John
Wilkins (1614-1672) in An essay towards a real character, and a
philosophical language [London: Printed for Sa. Gellibrand, and
for John Martyn, 1668]. He wrote: "Those Algebraical notions of
Absolute, Lineary, Quadratic, Cubic" (OED2)

QUADRATIC FORM. The term form perhaps in this sense was used by Euler in
“Novae
demonstrations circa divisors numerorum formae xx _ nyy” which was read
Nov. 20, 1775 at the St. Petersburg Academy and published after his death in Nova
acta acad. Petrop. I [1783], 1747 47-74. This paper is cited in Disquisitiones
Arithmeticae, article 151, page 104.

A translation is: “In this section we shall treat particularly of functions in two unknowns x, y of the form
axx + 2bxy + cyy
where a, b, c are given integers. We will call these functions forms of the second degree or simply forms.”

Gauss also used Latin terms which are translated binary, ternary, and quaternary forms.

A Google print search finds der binaren quadratischen Formen in a letter dated July 9, 1831, which may have been written by Gauss.

Some other uses in German of quadratic form can be found using Google print search, but these seem to refer to the term
as it is used elsewhere in number theory or in chemistry.

Quadratic form is found in English in 1859 in G. Salmon, Less. Mod.
Higher Alg.: "A quadratic form can be reduced in an infinity of
ways to a sum of squares, yet the number of positive and negative
squares in this sum is fixed" (OED2).

Binary quadratic form is found in English in 1861 in
“Report on the Theory of Numbers—Part III” by H. J. Stephen Smith [Google print search].

[James A. Landau]

The term QUADRATIC RESIDUE was introduced by Euler in a paper
of 1754-55 (Kline, page 611). The term non-residue is found
in a paper by Euler of 1758-59, but may occur earlier.

QUADRATRIX. The quadratrix of Hippias was probably invented by
Hippias but it became known as a quadratrix when Dinostratus used it
for the quadrature of a circle (DSB, article: "Dinostratus";
Webster’s New International Dictionary, 1909).

The term QUADRATRIX OF HIPPIAS was used by Proclus (DSB,
article: "Dinostratus").

The quadratrix of Hippias is the first named curve other than circle
and line, according to Xah Lee’s Visual Dictionary of Special Plane
Curves website.

QUADRATURE, QUADRATURE OF THE CIRCLE, and MECHANICAL QUADRATURE. The OED defines quadrature as the
“process of constructing geometrically a square equal in area to that of a
given figure, esp. a circle. More widely: the calculation of the area bounded
by, or lying under, a curve; the calculation of a definite integral, esp. by
numerical methods.”

The problem of squaring the circle was one of classical problems of Greek mathematics: see the MacTutor article
Squaring
the circle. The word quadrature appears in English in the 16th century: the earliest quotation in
the OED is from 1569 in Sanford’s translation of the
contemporary Latin work by Heinrich Corelius Agrippa Of the vanitie and uncertaintie of artes and sciences,
“Yet no Geometrician hath founde out the true
Quadrature [L. quadraturam] of the Circle.” In classical Latin the word
meant a division of land into squares.

Agrippa’s book was not a mathematics book and the problem of the quadrature of the circle was
well-known outside specialist mathematical circles: the OED’s earliest quotation using the phrase to square
the circle is from a 1624 sermon by John Donne
(1572-1631): “Goe not Thou about to Square eyther circle [sc. God or thyself].”

Other quotations in the OED illustrate the quadrature of other curves and the
connection with CALCULUS: “A method for
the Quadrature of Parabola’s of all degrees” from an obituary of Fermat in Philosophical
Transactions (1665-1678), 1, (1665 - 1666), 15-16 and “Drawing Tangents to Curves, finding their Curvatures,
their Lengths, and Quadratures” from W. Emerson’s The Doctrine of Fluxions 1743.

The term mechanical quadrature is found in German in F. G. Mehler, “Bemerkungen zur Theorie
der mechanischen Quadraturen,” J. Reine angew. Math63 (1864) 152-157.
By the early 20th century the term was established
in the English mathematical literature. A JSTOR search found G. D. Birkhoff’s “General Mean Value and Remainder Theorems with
Applications to Mechanical Differentiation and Quadrature,” Transactions of the American Mathematical
Society, 7, 1906), 107-136. The
methods of quadrature considered by Birkhoff went back to the early 18th century—one was due to Roger Cotes.

Mechanical quadrature did not usually involve a machine of any kind and in the 20th century the
term gave way to numerical quadrature and finally to numerical integration.
There is an early appearance of the latter term in the title A Course in Integration and
Numerical Integration (1915) by David Gibb. The term is used in the well-known
book of 1924, The Calculus of Observations by E. T. Whittaker & G.
Robinson. In the course of the 19th century mechanical integrating
machines or integrators were
developed by physicists. One is described in James Thomson’s “On an Integrating Machine
Having a New Kinematic Principle,” Proceedings of the
Royal Society of London, 24, (1875 - 1876), 262-265. The history
of these machines is related in “Integrators and Planimeters,” chapter 5 of H.
H. Goldstine The Computer from Pascal to von Neumann.

QUADRILATERAL appears in English in 1650 in Thomas Rudd’s
translation of Euclid.

The QUADRIVIUM of the European Middle Ages consisted of the four mathematical sciences, arithmetic,
geometry, astronomy and music. (According to T. L. Heath A History of Greek Mathematics, vol. 1 p.
12, the Pythagoreans divided MATHEMATICS in the same way.) The term was used by the Roman scholar
Boethius
in his Arithmetica and, according
to the DSB, this is “probably the
first time the word was used.” Boethius had a profound influence on education
in medieval Europe. Preceding the quadrivium in the syllabus was the trivium,
for which see the entry LOGIC. The word trivium is echoed in the
modern word TRIVIAL.

QUALITY CONTROL. In 1929 W. A. Shewhart gave an address to a meeting of the American
Association for the Advancement of Science on "Quality control of the manufactured
article." In 1930 Bell Telephone Labs. issued Shewhart’s "Economic quality control
of manufactured product". This 26 page pamphlet was followed the next year by
a 501 page book Economic Control of Quality of Manufactured Product. (JSTOR search)

The term QUANTIC for a homogeneous function of two or more variables with constant
coefficients was introduced by Arthur Cayley in “An Introductory Memoir on Quantics,”
Philosophical Transactions of the Royal Society of London,144 (1854). See
Collected Papers II p. 221.

QUANTIFIER (logic.) According to A. Church Introduction
to Mathematical Logic (1956, p. 288), "The notion of propositional function
and the use of quantifiers, originated with Frege in his Begriffsschrift
of 1879... The terms ‘quantifier’ and ‘quantification’ are Peirce’s." In 1885
C. S. Peirce wrote, "If the quantifying part, or Quantifier, contains Σx
and we wish to replace the x by a new index i, not already in
the Quantifier, and such that every x is an i, we can do so at
once by simply multiplying every letter of the Boolian having x as an
index by xi." Coll. Papers (1933) III. xiii. P. 232
(OED)

The termsuniversal quantifier and existential
quantifier became established in the 1930s. The OED gives a reference to
Łukasiewicz and Tarski writing in Polish in 1930. A JSTOR search
found universal quantifier in H. B. Curry "The Universal Quantifier in Combinatory Logic,"
Annals of Mathematics, 32, (1931), 154-180 and
existential quantifier in H. B. Curry "Apparent Variables From the Standpoint of Combinatory
Logic," Annals of Mathematics, 34, (1933), 381-404.

QUANTILE as a general term
covering quartiles, percentiles, etc. arrived about 60 years after the terms
it was generalising.

David (2001) gives M. G. Kendall’s "Note
on the Distribution of Quantiles for Large Samples," Supplement to the Journal
of the Royal Statistical Society, 7, (1940 - 1941), 83-85.

See QUARTILE.

QUARTILE. Higher and lower quartile are found in 1879 in Donald McAlister,
The Law
of the Geometric Mean, Proc. R. Soc. XXIX, p. 374:
"As these two measures, with the mean, divide the
curve of facility into four equal parts, I propose to call them the ’shigher
quartile’s and the ’slower quartile’s respectively. It will be seen that they correspond
to the ill-named ’sprobable errors’s of the ordinary theory" (OED2). McAlister
defined octiles on a similar principle. The topic of McAlister’s paper was suggested
by Francis Galton (see LOGNORMAL) and Galton may have had a hand in the terms.
However, McAlister’s words seems clear enough.

Upper and lower quartile
appear in 1882 in F. Galton, "Report of the Anthropometric Committee,"
Report of the 51st Meeting of the British Association for the Advancement
of Science, 1881, p. 245-260

(From David (1995) and Jeffrey K. Aronson "Francis Galton
and the Invention of Terms for Quantiles," Journal of Clinical Epidemiology,
54, (2001) 1191-1194.)

The term QUASI-PERIODIC FUNCTION was introduced by Ernest Esclangon (1876-1954) (DSB, article: Bohl).

QUATERNION (a group of four things) dates to the 14th century in English. The word appears in the King James
Bible (Acts 12:4), which refers to "four quaternions of soldiers."

See also HAMILTON, VECTOR and VECTOR ANALYSIS.
For quaternion terms that have not survived (in their original
meaning at least) see TENSOR and VERSOR.

QUEUEING. The OED2 shows a use of "a queueing system" and "a
complex queueing problem" in 1951 in David G. Kendall’s
"Some Problems in the Theory of Queues," Journal of the Royal Statistical Society,Series B, 13, 151-185 and a use of "queueing theory" in 1954 in Science News.
Kendall does not give a history of the "Theory of Queues"
but he says that, "the first major study of congestion problems was undertaken by
A. K. Erlang
in 1908 ... under the auspices of the Copenhagen Telephone Company."

[An interesting fact about the word queueing is that it contains five consecutive
vowels, the longest string of vowels in any English word, except for a few obscure
words not generally found in dictionaries.]

QUINDECAGON is found in English in 1570 in Henry Billingsley’s
translation of Euclid: "In a circle geuen to describe a quindecagon
or figure of fiftene angles" (OED2).

The OED2 shows one citation, from 1645, for pendecagon.

QUINTIC was used in English as an adjective in 1853 by
Sylvester in Philosophical Magazine: "May, To express the
number of distinct Quintic and Sextic invariants."

Quintic was used as a noun in 1856 by Cayley: "In the case of
a quantic of the fifth order or quintic" (from his Works,
1889) (OED2).

QUINTILE is found in 1922 in "The
Accuracy of the Plating Method of Estimating the
Density of Bacterial Populations", Annals of Applied Biology
by R. A. Fisher, H. G. Thornton, and W. A. Mackenzie: "Since the 3-plate
sets are relatively scanty, we can best test their agreement with theory by
dividing the theoretical distribution of 43 values at its quintiles, so that
the expectation is the same in each group." There are much earlier uses
of this term in astrology [James A. Landau].

QUOTIENT. Joannes de Muris (c. 1350) used numerus
quociens.

In the Rollandus Manuscript (1424) quotiens is used (Smith
vol. 2, page 131).

Pellos (1492) used quocient.

QUOTIENT (group theory). This term was introduced by
Hölder in 1889, according to a paper by Young in 1893.

Quotient appears in English in 1893 in a paper by Cayley,
"Note on the so-called quotient G/H in the theory of groups."

QUOTIENT GROUP. Otto Hölder (1859-1937) coined
the term factorgruppe. He used the term in 1889.

Quotient group is found in 1893 in Bull. N.Y. Math.
Soc. III. 74: "The quotient-group of any two consecutive groups
in the series of composition of any group is a simple group" (OED2).

Factor-group appears in English in G. L. Brown, "Note on
Hölder’s theorem Concerning the constancy of factor-groups,"
American M. S. Bull. (1895).

QUOTIENT RING is found in 1946 in Foundations of Algebraic Geometry
by Andre Weil: "Furthermore, every quotient-ring derived from this ring and a prime ideal in it
will be Noetherian."
[Google print search]